Implicit Function

Implicit Function

A function developed for differentiation of functions comprising variables but not easily written in y = f(x)  is called an implicit function (x). If x^2 + y^2 + 4xy + 25 = 0, then the dependent and independent variables “y” cannot be easily separated to describe it as a function of “y = f(x),” then it is an example of an implicit function (x). 

To better grasp implicit function, we need to understand it in a better manner. So let’s look at some real-world examples to better understand the concepts of implicit function and implicit function differentiation.

What is an Implicit Function?

It is possible to have an implicit function with numerous variables, one of which depends on another. Assuming the variables on one side of the equation are equalized to zero, we have an equation that expresses the function f(x, y). If y^2 + xy = 0, then it’s clear that there is an implicit function at work. In addition, an implicit function is one in which one variable is dependent on both the other two variables and the function f(x, y, z) = 0.

Explicit functions are those whose definition is as simple as y = f(x), where f is the function of x and y is the variable. In an implicit function, the x and y variables cannot be represented in the form y = f(x), and an implicit function has more than one solution for the given function. It is possible to write and express an implicit function expression as an equation with two or even more variables. The expression on the left-hand side of the equation comprises all variables and their respective constants and coefficients.

The relationship y = f(x) is a function that expresses y as a function of the input value x. As another example, g(x, y) = 0 depicts the relationship between x and y in an implicit function. Therefore, the value of x equals zero in the expression on the left-hand side when y is substituted. There are many examples of explicit functions, such as the phrases “x^2” and “y^2” being equal to “0,” and “ax^2 + bxy” being equivalent to “x^2” and “y^2” is equivalent to “0,” while “e^x + x – y + log y” being equivalent to “0” is an example of an implicit function.

Derivative of Implicit Function

Differentiation has been used to define the term “implicit function,” which refers to functions with many variables that are difficult to differentiate. The chain rule of differentiation of functions is used to determine implicit functions. 

To better comprehend implicit functions, let’s first take a closer look at explicit functions. For example, y = f(x) is a common formula for manipulating and expressing basic linear equations in x and y, and it is referred to as an explicit function. In this case, it’s easy to tell the difference between the dependent variable y and the independent variable x. 

To differentiate the implicit function, one must consider all of the variables in the equation, which can include more than one independent and dependent variable. We can isolate the expression from other variables by utilizing partial differentiation. 

Two easy steps are all that are required to differentiate an implicit function. When f(x, y) = 0 is decomposed into its components, the first step is to determine which two independent variables are zero. Then, algebraically shift the variables to determine the expression’s dy/dx  value. 

Steps to Compute the Derivative of an Implicit Function

• If y is a dependent variable and x is an independent variable in a given implicit function, then (or the other way around)
• Calculate the derivative of each term in the equation, taking into account the independent variable (it could be x or y) 
• The chain rule of differentiation must be used once we have differentiated
• If higher-order derivatives are required, solve the resulting equation for dy/dx  (or dx/dy  in the same way)

Properties of Implicit Function

It is useful to know the following properties to understand implicit functions better

• It is impossible to write y = f(x) for the implicit function f(x)
• When the function is implicit, it is always interpreted as f(x,y) = 0
• There are numerous variables involved in the implicit function
• The dependent and independent variables are used to formulate the implicit function 
• Several points are crossed by the vertical line drawn along the graph of an implicit function

Conclusion

An implicit function is an important topic to be studied to understand the deep concepts of calculus. Implicit functions have a vast scope and use in architecture and material science. To understand calculus, implicit functions are the building blocks. Through this topic, we will be able to understand the implicit functions.

Mathematicians use derivatives to express rates of change in calculus. Calculus is used extensively in various ways, including formulating a differential equation that includes an unknown function y=f(x) and its derivative. Sometimes, the solutions to these equations reveal how and why specific variables change.